gh-transformierte symmetrische Verteilungen

Tukey (1960) derived via the technique of transformation of variables starting from the normal distribution a family of skewed and leptokurtic distributions. Skewness and leptokurtosis are determined by two parameters g and h. Therefore, the family was called family gh-distributions. We modify Tukey's proposal such that other symmetric distributions than the normal distribution will be taken as starting point for the transformation of variables. We speak about a family "gh-transformed symmetrical distributions." Especially, we consider the Laplace distribution and the t-distribution with a fixed number of degrees. The aim is to show, what kind of substitution take place between a leptokurtic symmetric distribution and the parameter g and h. Because of numerical problems with maximum likelihood, Hoaglins (1983) technique of estimation by quantiles is use. We demonstrate how the three families of gh-transformed symmetrical distributions work for real financial data sets that stem from a skewed and leptokurtic distribution.